Transcript Absolute Stability with a Generalized Sector Condition Tingshu Hu Outline     Background, problems and tools Absolute stability with a conic sector, circle criterion, LMIs The generalized.

Absolute Stability with a
Generalized Sector Condition
Tingshu Hu
1
Outline




Background, problems and tools
Absolute stability with a conic sector, circle criterion, LMIs
The generalized sector bounded by PL functions (PL sector)
Composite quadratic Lyapunov functions
Main results: Estimation of DOA with invariant level sets
Quadratics : Invariant ellipsoid  LMIs
Composite quadratics : Invariant convex hull of ellipsoids  BMIs
An example
Building up the main results
─ Foundation: Stability analysis of systems with saturation
Main idea: Describing PL sector with saturation functions
Absolute stability  stability for a family of saturated systems
Summary
2
System with a conic sector condition
A system with a nonlinear and/or uncertain component:
u
v
 (u, t )
v
F (sI  A) 1 B
The conic sector condition:
k 2u
( (u, t )  k1u)( (u, t )  k2u)  0
k1u
u
Question: what is the
condition of robust stability for
all possible (u,t) satisfying
the sector condition?
3
Stability for a nonlinear system
Consider a nonlinear system:
x  f ( x, t )
Stability is about the convergence of the state to the origin
or an equilibrium point. Also, if it is initially close to the origin,
it will stay close.
Stability region: the set of initial x0 such that the state
trajectory converges to the origin.
Global stability: the stability region
is the whole state space.
x(t )  0 for all x0
4
Quadratic function and level sets
 Given a nn real symmetric matrix P, P=PT. If xTPx>0
for all xRn\{0}, we call P a positive definite matrix,
and denote P > 0. (Negative definite can be defined similarly)
 With P > 0, define V(x)= xTPx. Then V is a positive
definite function, i.e., V(x) > 0 for all xRn\{0}.
 Level sets of a quadratic function: Ellipsoids. Given r >0.
 ( P, r )  x  Rn : xT Px  r
 ( P, r1 )   ( P, r2 )   ( P, r3 )   ( P, r4 )
r3 r 4
r1 r 2
5
Quadratic stability
 The system: x  f ( x, t )
 Stability condition: If for all x  (P,r0 )\{0},
Px, x  xT Pf ( x, t )  0
(*)
Then  (P,r0 ) is a contractively invariant set and
a region quadratic stability.
 Condition (*) means that along
the boundary of  (P,r) for any
r  r0 , the vector x  f ( x, t )
points inward of the boundary
Px
x
In Lyapunov stability theory, the quadratic Lyapunov function is
6
replaced with a more general positive-definite function
Quadratic stability for linear systems
x  Ax
 Stability condition: If for all x  (P,r0 )\{0},
 For a linear system:
xT PAx  0
 xT ( AT P  PA) x  0
(*)
Then  (P,r0 ) is a contractively invariant set and
a region of quadratic stability.
 (*) is equivalent to
AT P  PA  0 (negative- definite)
 Lyapunov matrix inequality.
 As long as there exists a P satisfying
the matrix inequality, the linear system is stable
7
Absolute stability with conic sector
Consider again the system with a nonlinear component:
u
v
(u,t)
v
F(sI-A)-1B
The conic sector condition:
k 2u
( (u, t )  k1u)( (u, t )  k2u)  0
k1u
u
Absolute stability: the origin is
globally stable for any 
satisfying the sector condition
8
Conditions for absolute stability
Popov criterion
Circle criterion
LMI condition 
u
 (u, t )
v
Quadratic stability
v
k 2u
F (sI  A) 1 B
k1u
u
Description with linear differential inclusion (LDI):
x  ( A  kBF) x : k [k1 , k2 ]  co ( A  k1BF ) x, ( A  k2 BF ) x
Px
Quadratic stability: exists P=P’ >0 such that
( A  k1BF)' P  P( A k 1BF)  0
( A  k2 BF)' P  P( A k 2BF)  0
x
9
Motivation for a generalized sector

Limitations of the conic sector:
• not flexible
• could be too conservative
k 2u
k1u
u
Note: Subclass of the conic sector has
been considered, e.g., slope restricted,
Monotone ( Dewey & Jury, Haddad & Kapila,
Pearson & Gibson, Willems, Safonov et al,
Zames & Falb, etc.)
Our new approach
extend the linear boundary functions to nonlinear functions
basic consideration: numerical tractability
Our Choice: Piecewise linear convex/concave boundary functions
10
A piecewise linear (PL) sector
Let 1 and 2 be :
v
 1 (u)
 (u, t )
 2 (u)
 odd symmetric,
 piecewise linear
 convex or concave for u > 0
The generalized sector condition:
( (u, t )  1 (u))( (u, t )  2 (u))  0
Main feature: More flexible and
still tractable
u
v
 1 (u)
 2 (u)
u
11
A tool: the composite quadratic function
Given J positive definite matrices:
Denote
Q1 , Q2 ,, QJ > 0


    R J :  1    J  1,  j  0
The composite quadratic function is defined as:
Vc ( x) :
1
1
min x (  1Q1     J QJ  x
2  
The level set of VC is the
convex hull of ellipsoids
Convex, differentiable
12
Applying composite quadratics to conic sectors
Recall: A systems with conic sector condition can be described
with a LDI:
x coA1 x, A2 x
Theorem: Consider Vc composed from Qj’s. If there exist
i = 1,2 , j,k =1,2,…,J , such that
lijk ≥
0,
J
Ai Q j  Q j A   lijk (Qk  Q j )
'
i
k 1
Then Vc ( x)  0  x  0. ( V ( x)T x  0 )
Example: A linear difference inclusion: x(k+1)co{A1x, A2(a)x}
.4  0.4 ,
A1  0
0.4 0.4 
A2 (a )  00.4.4a  00..44/ a  , a  1


With quadratics, the maximal a ensuring stability is a1=4.676;
With composite quadratics (N=2), the maximal a is a2=7.546
13

Main results: Invariant level sets
Quadratics : Invariant ellipsoid
Composite quadratics :
Invariant convex hull of ellipsoids
An example
LMIs
BMIs
14
Absolute stability analysis via absolutely
invariant level sets
x  Ax  B ( Fx, t )
Consider the system:
For a Lyapunov candidate V ( x ), its 1-Level set is


LV (1)  x  Rn :V ( x)  1
The set LV (1) is contractively invariant (CI) if
V  V ( x)T ( Ax  B (Fx, t ))  0 x  LV (1) \ {0}, t  R
LV (1) is absolutely contractively invariant (ACI) if it is
contractively invariant for all   co {1, 2
 Quadratics : ACI ellipsoids,
 Composite quadratics: ACI convex hull of ellipsoids
15
Result 1: contractive invariance of ellipsoid
Consider the system,
k 0 u ,
k u  c u,
 1
1

(
u
)

x  Ax  B (Fx) ,


 k N u  c N u ,
if u  [0, b1 ]
if u  (b1 , b2 ]

if u  (bN , )
Theorem: An ellipsoid  ( Q 1 ) is contractively invariant iff
Q( A  k0 BF)'( A  k0 BF)Q  0
and there exist
Yi  R1n , i  1,2,...N
such that
QA' AQ  Yi ' B' BYi  0,

ci2
Yi  ki FQ

  0,
Q
Yi 'ki QF '

i  1,2, , N
16
Result 2: Quadratics → ACI ellipsoids
The system,
 k0 q u,
k u  c ,
x  Ax  B ( Fx, t ) ,
 1q
1q
 q (u )  
 (u, t )  co 1 (u ), 2 (u ),

 k N ,q u  c N ,q ,
u, t  R
if u  [0, b1q ]
if u  (b1q , b2 q ]

, q  1,2
if u  (bN ,q , )
Theorem: An ellipsoid  ( Q 1 ) is ACI if and only if
Q( A  k0q BF)'( A  k0q BF)Q  0,
q  1,2
and there exist Yi1, Yi 2  R1n , i  1,2,...N such that
QA' AQ  Yiq ' B' BYiq  0,

ciq2
Yiq  kiq FQ

  0,
Q
Yiq 'kiqQF '

i  1,2,, N , q  1,2
17
Result 3: ACI of convex hull of ellipsoids
Consider Vc composed from Qj’s. LVc (1) is the convex
hull of  (Qj-1).
Theorem: LVc (1) is ACI if there exist liqjk ≥ 0, i 0,1,…,N ,
q=1,2, j,k =1,2,…,J , such that
J
Q j ( A  k0 q BF )'( A  k0 q BF )Q j   l0 qjk (Qk  Q j ),
q  1,2
k 1
and there exist
Yi1 j , Yi 2 j  R1n , i [1, N ], j [1, J ], such that
J
Q j A' AQj  Yiqj ' B' BYiqj   liqjk (Qk  Q j ),
k 1

ciq2
Yiqj  kiq FQj 

  0,
Qj
Yiqj 'kiqQ j F '

i  [1, N ], q  1,2, j  [1, J ]
18
Optimizing ACI level sets
Choose reference points x1,x2,…,xK . Determine ACI LVc (1) such
that xp’s are inside LVc (1) with  maximized.
 x1
sup  ,
 1

x p

x p ' 
 x2
  0, p  [1, K ], j  [1, J ]
γ pj Q j 


j 1
J
J
Q j ( A  k0 q BF )'( A  k0 q BF )Q j   l0 qjk (Qk  Q j ),
q  1,2, j  [1, J ]
k 1
J
Q j A' AQj  Yiqj ' B' BYiqj   liqjk (Qk  Q j ),
k 1

ciq2
Yiqj  kiq FQj 

  0,
Qj
Yiqj 'kiqQ j F '

i  [1, N ], q  1,2, j  [1, J ]
19
Example
A second order system:


0.3 0.01
 0.6
A
, B
,


  1 0.3 
 0.7 
F  0.4  8.8
1
Reference point: x0   
1



1
x  Ax  B ( Fx),   co  , ,
Maximal : 0.8718

1.5
0.5
0
-0.5
-1
-1.5
-5
0
5
2
1.5
1
0.5
LVc(1):
(Q1-1):
(Q2
-1):
0
-0.5
-1
-1.5
-2
-2
20
-1.5
-1
-0.5
0
0.5
1
1.5
2
2
Composite quadratics + PL sector
max  0.8718
1.5
1
Quadratics + PL sector
max = 0.6401
0.5
0
-0.5
Quadratics + conic sector
max  0.4724
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
3
ACI convex hull
2
A closed-trajectory under
the “worst switching” w.r.t Vc
1
0
A diverging trajectory
-1
-2
-3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
21

Building up the main results
Stability analysis for systems with saturation
Describing PL sector with saturation functions
Stability for an array of saturated systems
Absolute stability
22
Stability analysis for systems with saturation
The system
x  Ax  b sat( f x) , sat(u)  sgn(u) min{|u |,1}
Problem: To characterize the (contractive) invariance of
 ( P)  x  Rn : x' Px  1
u
k1 u
Traditional approach: find k1, 0 < k1 ≤ 1, such that
(sat( fx)  k1 fx)(sat( fx)  fx)  0 x   ( P),| k1 fx | 1
umax
then use the traditional absolute stability analysis tools
Note: The condition takes form of bilinear matrix inequalities
23
New approach of dealing with saturation
The basic idea: If |v| ≤ 1, then
(sat(u)  v)(sat(u)  u)  1 u  R
v
1
u
For any row vector h,
(sat( fx)  hx)(sat( fx)  fx)  0  | hx | 1
1
Recall the traditional approach
(sat( fx)  k1 fx)(sat( fx)  fx)  0  | k1 fx | 1
We have full degree of freedom in choosing h as compared with
the one degree of freedom in choosing k1 in k1f.
Further more, the resulting condition for invariance
of ellipsoid
includes only LMIs
1
is necessary and sufficient
24
Foundation: The necessary and sufficient
condition for invariance of ellipsoid
Theorem: the ellipsoid  ( Q1) is contractively invariant for
x  Ax  b sat( f x) ,
if and only if there exists y  R1n , such that
Q ( A  bf )' ( A  bf )Q  0
QA' AQ  y ' b'by  0,
1 y 
 y ' Q   0,


25
Building-up tool: description of PL functions
with saturation functions
 (u )
k1
Consider a PL function with
only one bend
k0
 k 0u ,
 (u )  
k1u  c1u,
if u  [0, b1 ]
if u  (b1 , )
 k0  k1
 (u )  k1u  c1sat 
 c1

u 

b1
u
x  Ax  b ( f x) ,
 k0  k1 
x  ( A  k1b f ) x  c1b sat 
fx 
 c1

The necessary and sufficient condition for invariance
of ellipsoid follows.
26
Key step: description of PL functions with
saturation functions
A PL function,
k 0 u ,
k u  c u,

1
 (u )   1

 k N u  c N u ,
k 0u
 1 (u)
if u  [0, b1 ]
if u  (b1 , b2 ]

 2 (u)
 3 (u)
 (u )
if u  (bN , )


Define  i (u )  ki u  ci sat  k0  ki u 
 ci

u
Properties:
a)  (u )  min  i (u ) : j  1,2,, N  u  0
b)  i (u )  co  k0u, (u )
27
Putting things together:
Absolute stability via saturated systems
The original system and N systems with saturation,

S : x  Ax  B ( Fx, t ) ,  co   , 

 k0 q  kiq 
Siq : x  ( A  kiq B F ) x  ciq B sat 
Fx ,
 c

iq


i  [1, N ], q  1,2
ACI of a level set for S
CI of the level set for all Siq
Stability analysis results contained in:
T. Hu, Z. Lin, B. M. Chen, Automatica, pp.351-359, 2002
T. Hu and Z. Lin, IEEE Trans. AC-47, pp.164-169, 2002
T. Hu, Z. Lin, R. Goebel and A. R. Teel, CDC04, to be presented.
28
Summary
•
•
•
•
•
The systems:
Tool:
Problem:
Key step:
Main feature:
subject to PL sector condition
composite quadratic Lyapunov function
determine ACI sets (convex hull of ellipsoids)
description of PL functions with saturations
more flexible as compared with
conic sector, and still tractable
• Future topics: under PL sector condition,
characterize the nonlinear L2 gain
apply non-quadratics to study input-state,
input-output, state-output properties
29